A Kadison-Dubois representation for associative rings.
type: Item
instance of: scholarly article
title: A Kadison-Dubois representation for associative rings.
zbMATH Open document ID: 1086.16025
author: Igor Klep
published in: Journal of Pure and Applied Algebra
publication date: 2004-07-12T00:00:00Z
review text: The Kadison-Dubois representation has a long and remarkable history. It appeared in many settings and at different levels of generalization. Its generalization to the case of commutative rings was found by \textit{T. Jacobi} [his Dissertation paper and Math. Z. 237, No. 2, 259-273 (2001; Zbl 0988.54039)] and then extended by \textit{M. Marshall} [Math. Z. 242, No. 2, 217-225 (2002; Zbl 1060.13008)]. The aim of this paper is to generalize Jacobi's version of the Kadison-Dubois representation to associative rings. The author finds necessary and sufficient conditions for a module to satisfy the so called Kadison-Dubois property (KDP for short). For a preprime \(T\) and \(T\)-module \(M\) in a unital associative ring \(A\) he proves the equivalence of the following statements: (1) \(M\) satisfies the KDP, (2) \(M\) is Archimedean and its infinitesimal hull \(\text{Arch}(M)\) is closed under multiplication, (3) \(\text{Arch}(M)\) is an Archimedean preprime. This criterion is used to obtain the main result: if \(T\) is a preordering and \(M\) is an Archimedean \(T\)-module in a unital associative ring \(A\), then \(M\) satisfies the KDP.
reviewed by: Andrzej Sładek
Mathematics Subject Classification ID: 16W80
Mathematics Subject Classification ID: 13J30
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 06F25
Mathematics Subject Classification ID: 12D15
zbMATH DE Number: 2078859
zbMATH Keywords: Kadison-Dubois representations
zbMATH Keywords: preprimes
zbMATH Keywords: preprime modules
zbMATH Keywords: Archimedean modules
zbMATH Keywords: preorderings
MaRDI profile type: Publication
full work available at URL: j.jpaa.2003.10.035
OpenAlex ID: W2033276773
cites work: A Property of Ordered Rings
cites work: Summen n-ter Potenzen in Körpern.
cites work: Zum Darstellungssatz von Kadison-Dubois
cites work: Archimedean preorderings in non-commutative semialgebraic geometry
cites work: Complete precones on noncommutative integral domains
cites work: The real spectrum of a noncommutative ring
cites work: A note on David Harrison's theory of reprimes
cites work: Second note on David Harrison's theory of preprimes
cites work: A representation theorem for certain partially ordered commutative rings
cites work: Distinguished representations of strictly positive polynomials
cites work: A representation theory for commutative topological algebra
cites work: Anneaux preordonnes
cites work: Q5730197
cites work: A general representation theorem for partially ordered commutative rings
cites work: Higher level reduced Witt rings of skew fields
cites work: Holomorphy rings and higher level orders on skew fields
cites work: The moment problem for non-compact semialgebraic sets
cites work: Q2716044
cites work: Q4285035
cites work: The \(K\)-moment problem for compact semi-algebraic sets
cites work: Q4407585
DOI: 10.1016/J.JPAA.2003.10.035
description: scientific article; zbMATH DE number 2078859